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Tree Decompositions Meet Induced Matchings: Beyond Max Weight Independent Set

Authors Paloma T. Lima , Martin Milanič , Peter Muršič , Karolina Okrasa , Paweł Rzążewski , Kenny Štorgel

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Author Details

Paloma T. Lima
  • IT University of Copenhagen, Denmark
Martin Milanič
  • FAMNIT and IAM, University of Primorska, Koper, Slovenia
Peter Muršič
  • FAMNIT, University of Primorska, Koper, Slovenia
Karolina Okrasa
  • Warsaw University of Technology, Poland
Paweł Rzążewski
  • Warsaw University of Technology, Poland
  • University of Warsaw, Poland
Kenny Štorgel
  • Faculty of Information Studies, Novo mesto, Slovenia
  • FAMNIT, University of Primorska, Koper, Slovenia

Funding

  • Lima, Paloma T.: Supported by the Independent Research Fund Denmark grant agreement number 2098-00012B.
  • Milanič, Martin: Supported in part by the Slovenian Research and Innovation Agency (I0-0035, research program P1-0285 and research projects J1-3001, J1-3002, J1-3003, J1-4008, J1-4084, and N1-0370) and by the research program CogniCom (0013103) at the University of Primorska.
  • Muršič, Peter: Supported in part by the Slovenian Research and Innovation Agency (J1-4008).
  • Okrasa, Karolina: Supported by the Polish National Science Centre grant no. 2021/41/N/ST6/01507.
  • Rzążewski, Paweł: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme grant agreement number 948057.
  • Štorgel, Kenny: Supported in part by the Slovenian Research and Innovation Agency (research program P1-0383 and research projects J1-3002 and J1-4008).

Cite AsGet BibTex

Paloma T. Lima, Martin Milanič, Peter Muršič, Karolina Okrasa, Paweł Rzążewski, and Kenny Štorgel. Tree Decompositions Meet Induced Matchings: Beyond Max Weight Independent Set. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 85:1-85:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.85

Abstract

For a tree decomposition 𝒯 of a graph G, by μ(𝒯) we denote the size of a largest induced matching in G all of whose edges intersect one bag of 𝒯. The induced matching treewidth of a graph G is the minimum value of μ(𝒯) over all tree decompositions 𝒯 of G. Yolov [SODA 2018] proved that for graphs of bounded induced matching treewidth, tree decompositions with bounded μ(𝒯) can be computed in polynomial time and Max Weight Independent Set can be solved in polynomial time. In this paper we explore what other problems are tractable in such classes of graphs. As our main result, we give a polynomial-time algorithm for Min Weight Feedback Vertex Set. We also provide some positive results concerning packing induced subgraphs, which in particular imply a PTAS for the problem of finding a largest induced subgraph of bounded treewidth. These results suggest that in graphs of bounded induced matching treewidth, one could find in polynomial time a maximum-weight induced subgraph of bounded treewidth satisfying a given CMSO₂ formula. We conjecture that such a result indeed holds and prove it for graphs of bounded tree-independence number, which form a rich and important family of subclasses of graphs of bounded induced matching treewidth. We complement these algorithmic results with a number of complexity and structural results concerning induced matching treewidth, including a linear relation to treewidth for graphs with bounded degree.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • induced matching treewidth
  • tree-independence number
  • feedback vertex set
  • induced packing
  • algorithmic meta-theorem

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